Last episode was written by Nasu as a final epilogue once the game is finished.

"Last episode" can happen in different timelines/routes but it is sure to happen after Fate's true ending. The reason is that it is done as an epilogue for such ending (it actually says epilogue). Nasu has made epilogues for all 3 true endings. In the original game he did Heaven's Feel true ending epilogue. After some time he did Fate's true ending epilogue "last episode" and recently he's made UBW's true ending epilogue (an 11 page script that was handed to Ufotable in order to make the epilogue episode).

Some people say it is just fanservice or whatever but it is the way Nasu has wanted the true ending of the Fate route to end. The same with the other epilogues. I remember back in the day when Fate was released that haters were saying the true ending for UBW is supposed to be a bad ending to highlight Heaven's Feel happy ending.

Then Ufotable made a happy ending for it. They said it was Ufotable's fault.In the end it is based on a 11-page script written by Nasu which is twice as happy as the epilogue episode (the "it will all lead to our happily ever after, I just know it" or "my first priority is Shirou's happiness; all this was written by Nasu in his manuscript, which I read in japanese booklet II of the bluray edition). It shouldn't be too surprising though as future Rin from UBW had pretty much said the same in Ataraxia.

All 3 epilogues after each route are happy, it is the way Nasu has wanted it to be. If you make the right decisions you'll survive and you'll have a lovestruck girl at the end in a "happily ever after" manner. There's nothing weird about it, it always follows the same pattern."Last episode" can also happen in other timelines other than Fate true ending though.

Also, UBW Good ending has even more fanservice and has been severely criticised for that (even without mentioning how much fanservice it has, many say the laws of the Nasuverse shouldn't allow such ending with just 2 mages). Ataraxia's ending outside of the loop once 100% of the game is completed has lots of fanservice as well. Heaven's Feel True ending shouldn't exist, the normal way to end things is the normal ending.

Nasu has indeed written last episode, along with the rest of them, so it’s canon for his work and the canon conclusion to Fate's true ending as it's its epilogue (though in this case it can also happen in other timelines).

At the end of the day, FSN is mostly good and does more thing right than wrong, but it’s still deeply, deeply flawed, and endings in particular is not its strong side (and I'm talking about all happy endings if you reach the end of the game).